In order to approximate several roots of nonlinear equations, we presented a novel family of two-step optimal iterative methods in this study. The method is fourth-order convergent, requiring just four function evaluations each iteration, and it is optimal in terms of Kung-Traub's conjecture. We use complex dynamical analysis, often known as basins of attraction, to study local convergence and dynamical behavior. Numerical experiments on nonlinear problems in biomedical engineering are carried out to determine the method's efficiency and robustness in comparison to other methods. In terms of convergence rate, computational complexity, and stability, numerical findings show that the novel approach outperforms the well-known existing algorithms, especially for functions with higher multiplicities of order.

Accurate and robust image segmentation remains a fundamental challenge in computer vision, particularly in the presence of intensity inhomogeneity, noise, and weak object boundaries. To address these challenges, we propose a Robust Pythagorean Fuzzy Energy-Based Level Set (RPFELS) model, which integrates a novel fuzzy energy formulation with level set evolution to enhance segmentation precision and resilience against noise. The model introduces a Pythagorean fuzzy divergence term to refine energy optimization, ensuring adaptive boundary preservation and reducing sensitivity to intensity variations. Additionally, a bounded fuzzy energy constraint is incorporated to ensure numerical stability and prevent energy leakage during evolution. Extensive experiments on benchmark datasets, including medical and natural images, validate the effectiveness of RPFELS. The model consistently outperforms recent selective segmentation methods in terms of Dice Score, Jaccard Index, and Hausdorff Distance, achieving superior segmentation accuracy and reduced boundary errors. Furthermore, a detailed statistical significance analysis using paired t-tests confirms that the observed improvements are statistically significant (p-value $<$ 0.01), reinforcing the reliability of the proposed approach. Moreover, RPFELS exhibits higher computational efficiency, achieving faster convergence rates compared to existing methods. These findings highlight the robustness and versatility of the proposed approach in handling challenging segmentation scenarios, making it suitable for applications in medical imaging, remote sensing, and industrial defect detection. By ensuring bounded energy evolution and statistically validated performance gains, our model sets a new benchmark in selective segmentation.

Neutrosophy is a special area of philosophy that explains the nature, genesis and scope of neutralities, like the interactions with diverse ideational hues. It showed the degree of indeterminacy as an independent component that was the extension of an intuitionistic set. In this paper, the interpretation of the linear equation of type $\mathcal{A}\mathcal{X} +\mathcal{B} =\mathcal{C}$ are discussed in a neutrosophic environment. It is observed that the equations $\mathcal{A}\mathcal{X} +\mathcal{B} =\mathcal{C}$, $\mathcal{A}\mathcal{X} =\mathcal{C} -\mathcal{B}$ and $\mathcal{A}\mathcal{X} -\mathcal{C} =-\mathcal{B}$ are same and their solution are also same in crisp sense. But, in the neutrosophic sense, the solutions to the above equations are different. Mathematical operations on intervals are considered for the purpose of solution and analysis. Further, an application of budgeting-financing is described with the help of neutrosophic fuzzy equation.

Automobiles play a vital role in daily life, providing suitable and efficient transportation for work, school, and errands. They also support essential services like emergency response, goods delivery, and public transportation systems. This increased variety means that car manufacturers are competing intensely to attract customers and maximize their profits. However, making the right choice when buying a car can be challenging due to the wide range of factors to consider. This study introduces a new approach that uses Dombi operators combined with T-spherical fuzzy numbers (T-SFNs) to help improve the decision-making process. This method reduces the uncertainty and imprecision that often comes with decision-making, especially when selecting a car. The aim is to help customers make better, more informed choices and avoid financial difficulties. To achieve this, the study develops several innovative operators namely T-spherical fuzzy Dombi weighted averaging (T-SFDWA), T-spherical fuzzy Dombi ordered weighted averaging (T-SFDOWA), T-spherical fuzzy Dombi weighted geometric (T-SFDWG), T-spherical fuzzy Dombi ordered weighted geometric (T-SFDOWG). These methods offer flexibility, suppleness and can adapt to real-world problems where factors are constantly changing. By managing uncertainty and hesitation effectively, these approaches help decision-makers evaluate complex situations with multiple variables. A practical example, such as choosing a car, demonstrates how these approaches can evaluate important criteria like price, safety, and fuel efficiency. Ultimately, these methods ensure that consumers can make the best decision, even in uncertain and complex situations.